Method and Apparatus for Broadband Modeling of Current Flow in Three-Dimensional Wires of Integrated Circuits

ABSTRACT

A new surface impedance model for extraction in lossy two-dimensional (2D) interconnects of rectangular cross-section is presented. The model is derived directly from the volumetric electric field integral equation (EFiE) under the approximation of the unknown volumetric current density as a product of the exponential factor describing the skin-effect and the unknown surface current density on the conductor&#39;s periphery. By proper accounting for the coupling between the boundary elements situated on the top and bottom surfaces of conductor with the elements located on the side-walls, the model maintains accuracy from DC to multi-GHz frequencies as well as for conductors with both large and small thickness/width ratios. A generalization of the full-periphery surface impedance model to the three-dimensional electric field integral equation is also described.

This application claims the benefit under 35 U.S.C. 119(e) of U.S.provisional application Ser. No. 61/041,262, filed Apr. 1, 2008.

FIELD OF THE INVENTION

The present invention relates to a method of modelling surface impedanceof a conductor having a current flow, and furthermore relates to aresulting surface impedance model and its implementation using asuitable method of moment discretization scheme. The surface impedancemodel is derived directly from the volumetric electric field integralequation under the approximation of the unknown volumetric currentdensity as a product of the exponential factor describing theskin-effect and the unknown surface current density on the conductor'speriphery.

BACKGROUND

Rigorous and efficient electromagnetic modeling of interconnects can beachieved through incorporation of appropriate surface impedance modelsinto the boundary-element discretization techniques such as MoM. To beuseful the model must be accurate in the broad range of frequenciesspanning from DC to tens of GHz, applicable to the conductorcross-sections with both small and large thickness-to-width ratios,computationally efficient, and also easy to retrofit into existing MoMsolvers.

A variety of surface impedance models have been proposed in the past asdescribed in the following documents:

1. J. D. Morsey, et. al., “Finite-Thickness Conductor Models forFull-Wave Analysis of Interconnects With a Fast Integral EquationMethod,” IEEE Trans. on Advanced Packaging, vol. 27, no. I, pp. 24-33,February 2004.

2. J. Rautio, et al., “Microstrip Conductor Loss Models forElectromagnetic Analysis,” IEEE Trans. on Microwave Theory Tech., vol.51, no. 3, pp. 915-921, March 2003.

3. F. Ling, et. al., “Large-Scale Broad-Band Parasitic Extraction forFast Layout Verification of 3-D RF and Mixed-Signal On-Chip Structures,”IEEE Trans. Microwave Theory Tech., vol. 53, no. 1, pp. 264-273, January2005.

4. A. W. Glisson, “Electromagnetic scattering by arbitrarily shapedsurfaces with impedance boundary conditions,” Radio Science, vol. 27,no. 6, pp. 935-943, November-December 1992,

The above mentioned requirements have not been satisfactorily met thusfar by the above noted documents and this consequently motivates theon-going quest for more efficient and accurate models.

Other prior art references relevant to the present invention include thefollowing.

5. C. R. Paul, Analysis of Mulliconduclor Transmission Lines, Ch. 3,John Wiley & Sons, Inc., Toronto, CA, 1994.

6. D. DeZutter, et al., “Skin Effect Modeling Based on a DifferentialSurface Admittance Operator,” IEEE Trans. Microwave Theory Tech., vol.53, no. 8, pp. 2526-2538, August. 2005.

7. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagneticscattering by surfaces of arbitrary shapes,” IEEE Trans. AntennasPropag., Vol. 30, pp. 409-418, May 1982.

8. K. A. Michalski and D. Zheng, “Electromagnetic scattering andradiation by surfaces of arbitrary shape in layered media, Part I:Theory,” IEEE Trans. Antennas Propag., Vol. 38, pp. 335-344, March 1990.

Any documents referred to in the accompanying specification are herebyincorporated by reference.

SUMMARY OF THE INVENTION

In this document a surface impedance model is derived as a result ofvolumetric electric field integral equation (EFIE) reduction to thesurface EFIE, via approximation of the unknown cross-sectionalvolumetric current as a product of the exponential factor describing theskin-effect and the surface current density on the periphery of theconductor. Thus, the derived surface EFIE allows for both theflexibility in current variation on the conductor's surface, which isessential for adequate capturing of the proximity effects, and theproper exponential attenuation of the current inside of the conductor,which is critical for accurate skin-effect description. The numericallyextracted per-unit-length (p.u.l) resistance and inductance matricesdemonstrate the model to be accurate in a wide range of frequencies, aswell as for both large and small thickness-to-width cross-sectionalaspect ratios.

According to one aspect of the invention there is provided a method ofmodelling surface impedance of a conductor having a current flow, themethod comprising:

formulating a volumetric electric field integral equation with respectto an unknown volumetric current density of the current flow in theconductor;

representing the unknown volumetric current density in the volumetricelectric field integral equation as a product of a current density of aperipheral surface of the conductor and an exponential factor describinga cross-sectional distribution of the current according to skin effect;

adopting an approximation of Green's function across a cross-section ofthe conductor; and

reducing the volumetric electric field integral equation including theunknown volumetric current density representation to a surface integralequation using the approximation of Green's function.

The method preferably includes adopting an approximation of Green'sfunction by fixing Green's function across the cross-section of theconductor.

The method is suitable for conductors having either a rectangularcross-section or a polygonal, non-rectangular cross-section.

The volumetric electric field integral equation is preferably formulatedwith respect to the unknown volumetric current density j_(z) in across-section of the conductor due to a vector of excitation V_(p.u.l.)by enforcing Ohm's law E_(z)(ρ)=σ⁻¹j_(z)(ρ) inside the conductor (ρ∈S)as follows:

σ⁻¹j_(z)(ρ) + ωμ₀∫_(S)G(ρρ^(′))j_(z)(ρ^(′))S^(′) = −V_(p.u.l.)(ρ).

For a conductor which extends in a z-direction and has a rectangularcross-section, the method preferably includes approximating the unknownvolumetric current density, represented as j_(z), across the conductoraccording to skin-effect of a plane-wave incident on a conducting planewith infinite extension as follows:

${j_{z}(\rho)} \cong {\frac{\; {k_{\sigma}\left( {{{J_{z}^{t}(y)}^{{- }\; {k_{\sigma}{({X - x})}}}} + {{J_{z}^{b}(y)}^{{- }\; k_{\sigma}x}}} \right)}}{1 - ^{{- }\; k_{\sigma}X}} + \frac{\; {k_{\sigma}\left( {{{J_{z}^{l}(x)}^{{- }\; k_{\sigma}y}} + {{J_{z}^{r}(x)}^{{- }\; {k_{\sigma}{({Y - y})}}}}} \right)}}{1 - ^{{- }\; k_{\sigma}Y}}}$

where J_(z) ^(t)(y), J_(z) ^(b)(y), J_(z) ^(l)(x), J_(z) ^(r)(x), areunknown surface current densities at points of radius-vector ρprojections onto respective top, bottom, left, and right walls of theconductor.

Reducing the volumetric electric field integral equation to a surfaceelectric field integral equation with respect to the current density ofthe peripheral surface of the conductor is preferably accomplished byrestricting an observation point ρ to a periphery of the conductor asfollows:

Z^(σ){J_(z)(ρ)} + ωμ₀∫_(C)G(ρρ^(′))J_(z)(ρ^(′))c^(′) = −V_(p.u.l.)(ρ)

The method preferably further includes combining the unknown surfacecurrent densities over the top, bottom, left and right sides andevaluating integrals thereof over a conductor thickness T and conductorwidth W as follows:

${\int_{V}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {j\left( r^{\prime} \right)}}{v^{\prime}}}} \cong {\int_{S}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {J\left( r^{\prime} \right)}}{{s^{\prime}}.}}}$

The volumetric electric field integral equation may be formulated withrespect to the unknown volumetric current density of the current flow inthe conductor for current flows along more than one axis for threedimensional current flows.

The unknown volumetric current density may be related to unknown surfacecurrent densities for each surface of the conductor in which eachunknown surface current density is represented as a multidimensionalvector.

The method may further include reducing the volumetric electric fieldintegral equation to the surface integral equation in which a tangentialelectrical field at any point on a surface of the conductor is relatedto a surface current density at corresponding points on all segments ofa cross-sectional periphery of the conductor and/or the tangentialelectrical field at any point on a surface of the conductor is relatedto a tangential magnetic field at corresponding points on all segmentsof a cross-sectional periphery of the conductor.

According to a second aspect of the present invention there is provideda surface impedance model for a conductor having a current flow, themodel comprising:

a surface electric field integral equation resulting from a reduction ofa volumetric electric field integral equation in which an unknowncross-sectional volumetric current of the volumetric electric fieldintegral equation is approximated as a product of an exponential factordescribing a cross-sectional distribution of the current according toskin effect and a current density on a peripheral surface of theconductor.

According to a further aspect of the present invention there is provideda method of implementing a surface impedance model of a conductorcomprising a surface electric field integral equation including asurface impedance term and an integral term resulting from a reductionof a volumetric electric field integral equation in which an unknowncross-sectional volumetric current of the volumetric electric fieldintegral equation is approximated as a product of an exponential factordescribing a cross-sectional distribution of the current according toskin effect and a current density on a peripheral surface of theconductor, the method including:

discretizing the surface electric field integral equation of a peripheryof the conductor into a discrete form comprising a sum of a sparsematrix corresponding to the surface impedance term of the surfaceelectric field integral equation and a dense matrix corresponding to theintegral term of the surface electric field integral equation; and

identifying for each discrete element of the discrete form, otherdiscrete elements which are related via the surface impedance model.

Preferably the method comprises using the surface impedance model inconjunction with a suitable method of moment discretization scheme.

The dense matrix may be represented as follows:

Z _(mn) ^(A) =iωμ ₀∫_(ΔC) _(n) G(ρ_(m)|ρ′)dc′.

According to the present invention as described herein, a simple andeffective surface impedance model suitable for RL-extraction in 2Dinterconnects of rectangular cross-section with both small and largethickness/width ratio is demonstrated. The model maintains accuracy fromDC to multi-GHz frequencies due to properly captured skin-effect currentattenuation off the conductor cross-sectional periphery. The extensionof the proposed model to 3D Rao-Wilton-Glisson MoM discretization ofEFIE is also demonstrated.

Some embodiments of the invention will now be described in conjunctionwith the accompanying drawings in which:

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 (a) is a schematic representation of an interconnect geometryrelating to the volumetric current density distribution across theconductor approximated according to skin effect.

FIG. 1 (b) is a schematic representation of the interconnect of FIG. 1(a) discretized with N=14 piece-wise basis functions.

FIG. 1 (c) is a schematic representation of additional matrix fillingsdue to the application of a full-peripheral Z_(s) model according to thepresent invention.

FIG. 2 illustrates graphical representations of inductance andresistance computations relative to frequency for an isolated squaresignal line with σ=57.2 MS/m.

FIG. 3 illustrates graphical representations of inductance andresistance computations relative to frequency for a differential linepair of conductors with σ=56 MS/m.

FIG. 4 is a schematic representation of exemplary source triangles onthe left, top, and bottom walls yielding non-zero ramp function to rampfunction interactions with an observation triangle on the top wall inthe discretize surface impedance operator.

FIG. 5 is a schematic representation of an example of an mth sourcetriangle on the right wall and mth observation triangle on the top-wallyielding non-zero ramp function to ramp function interactions in thediscretize surface impedance operator.

In the drawings like characters of reference indicate correspondingparts in the different figures.

DETAILED DESCRIPTION

The formulation of a surface impedance model as described hereininitially considers a generic 2-D interconnect structure ofcross-section S and conductivity σ situated in homogeneous nonmagneticmedium of complex relative permittivity {circumflex over (∈)}=∈+σ/(iω∈)where ω represents the angular frequency and i=√{square root over (−1)}.The interconnect is made up of a very good conducting material (i.e.σ>>ω∈). Assuming the current flow in the z-direction, we relate theelectric field E_(z) to volumetric current density j_(z) through thevector potential A_(z) as follows:

$\begin{matrix}{{E_{z}(\rho)} = {{{- {\omega}}\; {A_{z}(\rho)}} - \frac{{\varphi (\rho)}}{z}}} & (1)\end{matrix}$

where Φ is the scalar potential, and ρ=x{circumflex over (x)}+yŷ is the2-D position vector depicted in the coordinate system ρ(t,w), as shownin FIG. 1( a) and {circumflex over (z)}={circumflex over (x)}×ŷ, and

$\begin{matrix}{{A_{z}(\rho)} = {\mu_{0}{\int_{S}{{G\left( {\rho \rho^{\prime}} \right)}{j_{z}\left( \rho^{\prime} \right)}{S^{\prime}}}}}} & (2)\end{matrix}$

In equation (2), G is the two-dimensional free-space quasi-staticGreen's function

G(ρ|ρ′)=G(x,y|x′,y′)=−1/(2π)ln(|ρ−ρ′|)  (3)

where the logarithmic nature of the Green's function is obtained due toconsidering only the dominant conduction current σE_(z) and dropping offthe negligible displacement current ω∈E_(z). The volumetric EFIE withrespect to unknown current j_(z) in the conductor cross-section due toper-unit-length excitation follows from Ohm's law E_(z)(ρ)=σ⁻¹j_(z)(ρ)enforced inside the conductor (ρ∈S)

$\begin{matrix}{{{\sigma^{- 1}{j_{z}(\rho)}} + {{\omega\mu}_{0}{\int_{S}{{G\left( {\rho \rho^{\prime}} \right)}{j_{z}\left( \rho^{\prime} \right)}{S^{\prime}}}}}} = {- {V_{p.u.l.}(\rho)}}} & (4)\end{matrix}$

The boundary-element method such as MoM seeks to decrease thecomputational complexity of the numerical solution in volumetric EFIE(4) through its reduction to a surface EFIE where the unknown functionis the current distribution J_(z)(ρ) on the conductor surface ρ∈∂S. Inorder to perform such reduction let us consider a rectangularcross-section wire with focal coordinate system shown in FIG. 1( a). Thesought volumetric current density J_(z) distribution across theconductor can be approximated according to skin-effect as follows:

$\begin{matrix}{{j_{z}(\rho)} \cong {\frac{\; {k_{\sigma}\left( {{{J_{z}^{t}(y)}^{{- }\; {k_{\sigma}{({X - x})}}}} + {{J_{z}^{b}(y)}^{{- }\; k_{\sigma}x}}} \right)}}{1 - ^{{- }\; k_{\sigma}X}} + \frac{\; {k_{\sigma}\left( {{{J_{z}^{l}(x)}^{{- }\; k_{\sigma}y}} + {{J_{z}^{r}(x)}^{{- }\; {k_{\sigma}{({Y - y})}}}}} \right)}}{1 - ^{{- }\; k_{\sigma}Y}}}} & (5)\end{matrix}$

where J_(z) ^(t)(y), J_(z) ^(b)(y), J_(z) ^(l)(x), J_(z) ^(r)(x) are theunknown surface current densities at the points of radius-vector Pprojections onto the top, bottom, left, and right walls of theconductor, respectively as shown in FIG. 1( a). In Equation (5)k_(σ)=√{square root over (ωμ₀σ/2)}(1−i) is the complex wavenumber of theconductor V material, while X and Y denote the cross-sectional thicknessand width, respectively. Substitution of formula (5) into volumetricEFIE (4) and ignoring variation of the Green's function along thecoordinate normal to the conductor's periphery yields:

$\begin{matrix}{{\int_{S}{{G\left( {\rho \rho^{\prime}} \right)}{j_{z}\left( \rho^{\prime} \right)}{S^{\prime}}}} \cong {{\int_{C^{t}}{{G\left( {{\rho X},y^{\prime}} \right)}{J_{z}^{t}\left( y^{\prime} \right)}{y^{\prime}}}} + {\int_{C^{b}}{{G\left( {{\rho 0},y^{\prime}} \right)}{J_{z}^{b}\left( y^{\prime} \right)}{y^{\prime}}}} + {\int_{C^{l}}{{G\left( {{\rho x^{\prime}},0} \right)}{J_{z}^{l}\left( x^{\prime} \right)}{x^{\prime}}}} + {\int_{C^{r}}{{G\left( {{\rho x^{\prime}},Y} \right)}{J_{z}^{r}\left( x^{\prime} \right)}{x^{\prime}}}}}} & (6)\end{matrix}$

Note that ignoring Green's function variation along the normal to theconductor surface is optional. Instead, one may choose to include effectof Green's function variation by performing integration along thiscoordinate numerically to given precision.

Restriction of the observation point ρ to the conductor periphery C inequation (4) together with the approximation in equation (6) reduces thevolumetric EFIE (4) to the surface EFIE with respect to the unknownsurface current density J_(z)(ρ′) as follows:

$\begin{matrix}{{{Z^{\sigma}\left\{ {J_{z}(\rho)} \right\}} + {{\omega\mu}_{0}{\int_{C}{{G\left( {\rho \rho^{\prime}} \right)}{J_{z}\left( \rho^{\prime} \right)}{c^{\prime}}}}}} = {- {V_{p.u.l.}(\rho)}}} & (7)\end{matrix}$

where the radius-vectors ρ and ρ′ reside on the conductor surface C=∂S,C being the union of the top, bottom, left, and right conductor sidesC^(t), C^(b), C^(l), and C^(r) respectively.

In EFIE (7) Z^(σ){J_(z)(ρ)} denotes the surface impedance operatorrelating z-directed electric field on the surface of conductor E_(z)(ρ)to the values of the surface current densities J_(z)(ρ) at thecorresponding points on the top, bottom, left, and right sides of theconductor (FIG. 1 a) according to Ohm's law and the approximation inequation (5).

$\begin{matrix}{{Z^{\sigma}\left\{ {J_{z}(\rho)} \right\}} = {\left\lbrack {{E_{z}^{t}(y)},{E_{z}^{b}(y)},{E_{z}^{l}(x)},{E_{z}^{r}(x)}} \right\rbrack^{T} = {\frac{\; k_{\sigma}}{\sigma} \times {\begin{bmatrix}\frac{1}{1 - ^{{- }\; k_{\sigma}X}} & \frac{^{{- }\; k_{\sigma}X}}{1 - ^{{- }\; k_{\sigma}X}} & \frac{^{{- }\; k_{\sigma \; y}}}{1 - ^{{- }\; k_{\sigma}Y}} & \frac{^{{- }\; k_{\sigma \; {({Y - y})}}}}{1 - ^{{- }\; k_{\sigma}Y}} \\\frac{^{{- }\; k_{\sigma}X}}{1 - ^{{- }\; k_{\sigma}X}} & \frac{1}{1 - ^{{- }\; k_{\sigma}X}} & \frac{^{{- }\; k_{\sigma \; y}}}{1 - ^{{- }\; k_{\sigma}Y}} & \frac{^{{- }\; k_{\sigma \; {({Y - y})}}}}{1 - ^{{- }\; k_{\sigma}Y}} \\\frac{^{{- }\; {k_{\sigma}{({X - x})}}}}{1 - ^{{- }\; k_{\sigma}X}} & \frac{^{{- }\; k_{\sigma}x}}{1 - ^{{- }\; k_{\sigma}X}} & \frac{1}{1 - ^{{- }\; k_{\sigma}Y}} & \frac{^{{- }\; k_{\sigma \; Y}}}{1 - ^{{- }\; k_{\sigma}Y}} \\\frac{^{{- }\; {k_{\sigma}{({X - x})}}}}{1 - ^{{- }\; k_{\sigma}X}} & \frac{^{{- }\; k_{\sigma}x}}{1 - ^{{- }\; k_{\sigma}X}} & \frac{^{{- }\; k_{\sigma \; Y}}}{1 - ^{{- }\; k_{\sigma}Y}} & \frac{1}{1 - ^{{- }\; k_{\sigma}Y}}\end{bmatrix} \cdot {\left\lbrack {{J_{z}^{t}(y)},{J_{z}^{b}(y)},{J_{z}^{l}(x)},{J_{z}^{r}(x)}} \right\rbrack^{T}.}}}}} & (8)\end{matrix}$

The operator behavior at dc is examined via taking the limit of equation(8) as it approaches zero. Using Taylor approximation for very smallexponential arguments, i.e. e^(a)≅a+1,

$\begin{matrix}{{\lim\limits_{\omega\rightarrow 0}{Z^{\sigma}\left\{ {J_{z}(\rho)} \right\}}} = {\begin{bmatrix}{{1/\sigma}\; X} & {{1/\sigma}\; X} & {{1/\sigma}\; Y} & {{1/\sigma}\; Y} \\{{1/\sigma}\; X} & {{1/\sigma}\; X} & {{1/\sigma}\; Y} & {{1/\sigma}\; Y} \\{{1/\sigma}\; X} & {{1/\sigma}\; X} & {{1/\sigma}\; Y} & {{1/\sigma}\; Y} \\{{1/\sigma}\; X} & {{1/\sigma}\; X} & {{1/\sigma}\; Y} & {{1/\sigma}\; Y}\end{bmatrix} \cdot \begin{bmatrix}{J_{z}^{t}(y)} \\{J_{z}^{b}(y)} \\{J_{z}^{l}(x)} \\{J_{z}^{r}(x)}\end{bmatrix}}} & (9)\end{matrix}$

which indicates the dc resistance seen by the four surface currentdensities on the conductor's periphery. At high frequencies, theexponential terms of the operator become negligible, thus, turning theoperator's off-diagonal elements to zero whereas the diagonal terms tendto (1+i) √{square root over (ωμ₀/2σ)}, which corresponds to the localsurface impedance of the half-space with conductivity σ. From equation(8) it is also observed that lim_(σ→+∞)Z^(σ){J_(z)(ρ)}=0, which impliesthat equation (7) becomes an EFIE for a perfectly conductinginterconnect.

Implementation of Method of Moments

The proposed surface impedance model is intended for use in conjunctionwith an appropriate MoM discretization scheme. Below we demonstrate theMoM implementation with N=14 piece-wise basis functions distributed overthe conductor periphery as shown in FIG. 1( b). Under the abovediscretization the surface EFIE (7) is reduced to the linear algebraicequations as follows:

(Z ^(σ+) Z ^(A))·J=V  (10)

where Z_(mn) ^(A)=iωμ₀∫_(ΔC) _(n) G(ρ_(m)|ρ′)dc′ is the dense matrix ofthe vector potential interactions, V_(m)=−V_(p.u.l.)(ρ_(m)) is thevector of excitation, J_(m)=J_(z)(ρ_(m)) is the vector of unknowncurrent densities on the conductor's periphery, and Z_(mn) ^(σ) is thesparse matrix corresponding to the surface impedance operator inequation (8). Indexes m and n in the above matrices run from 1 to N. Thenon-zero elements in the sparse matrix Z^(σ) for this particulardiscretization are shown in FIG. 1( c) with the patterned squares. Theblack squares in FIG. 1( c) depict the non-zero entries in Z^(σ)corresponding to the field E_(z) tested at the observation pointρ={circumflex over (x)}X+ŷy₃ situated in the middle of the 3rd elementon the top peripheral segment C^(t). In accord with equation (8), thisfield is related to the surface current density on the top, bottom andside walls as

$\begin{matrix}{{E_{z}^{t}\left( y_{3} \right)} = {{Z_{3,3}^{\sigma,{tt}}{J_{z}^{t}\left( y_{3} \right)}} + {Z_{3,3}^{\sigma,{tb}}{J_{z}^{b}\left( y_{3} \right)}} + {Z_{3,1}^{\sigma,{tl}}{J_{z}^{l}\left( x_{1} \right)}} + {Z_{3,1}^{\sigma,{tr}}{J_{z}^{r}\left( x_{1} \right)}}}} & (11)\end{matrix}$

where the matrix Z^(σ) entries are

$\begin{matrix}{{Z_{3,3}^{\sigma,{tt}} = {\frac{\; k_{\sigma}}{\sigma}\frac{1}{1 - ^{{- }\; k_{\sigma}X}}}},\mspace{14mu} {Z_{3,3}^{\sigma,{tb}} = {\frac{\; k_{\sigma}}{\sigma}\frac{^{{- }\; k_{\sigma}X}}{1 - ^{{- }\; k_{\sigma}X}}}},{Z_{3,1}^{\sigma,{tl}} = {\frac{\; k_{\sigma}}{\sigma}\frac{^{{- }\; k_{\sigma}y_{3}}}{1 - ^{{- }\; k_{\sigma}Y}}}},\mspace{14mu} {Z_{3,1}^{\sigma,{tr}} = {\frac{\; k_{\sigma}}{\sigma}{\frac{^{{- }\; {k_{\sigma}{({Y - y_{3}})}}}}{1 - ^{{- }\; k_{\sigma}Y}}.}}}} & (12)\end{matrix}$

The remaining non-zero element in the surface impedance matrix Z^(σ) arefilled out in a similar manner.

Numerical Results

The proposed full-peripheral surface impedance model has been tested forseveral interconnects with both large and small thickness/width ratiosand has shown a reliable performance. A detailed description of p.u.l.resistance and inductance matrices for k conductor system can be foundin C. R. Paul, Analysis of Mulliconduclor Transmission Lines, Ch. 3,John Wiley & Sons, Inc., Toronto, CA, 1994.

In the first numerical experiment, the p.u.l. resistance and inductanceare extracted for the 4.62 mm-wide copper wire of square cross-sectionrepresented in FIG. 2 via the solution of EFIE (7) with surfaceimpedance equation (8). The extracted parameters are compared to theaccurate volumetric EFIE solution equation (4) as well as to theextracted p.u.l. resistance presented in D. DeZutter, et al., “SkinEffect Modeling Based on a Differential Surface Admittance Operator,”IEEE Trans. Microwave Theory Tech., vol. 53, no. 8, pp. 2526-2538,August. 2005. The MoM discretization at all frequencies consisted of 20equidistant segments per each side of the conductor. The results fromthe two-plane surface impedance model utilized in the 2.5D interconnectmodels, as described in J. D. Morsey, et. al., “Finite-ThicknessConductor Models for Full-Wave Analysis of Interconnects With a FastIntegral Equation Method,” IEEE Trans. on Advanced Packaging, vol. 27,no. 1, pp. 24-33, February 2004, are also shown in FIG. 2. The error inthe latter increases at high frequencies because only the top and bottomsegments of the conductor's periphery support the current. Such modelbecomes inaccurate when the thickness of conductor is comparable to itswidth. The EM analysis yielded a dc resistance of 0.8191 mΩ/m and the√{square root over (ƒ)} resistance dependence at high frequency isapparent from FIG. 2. The p.u.l. inductance is undefined under thisscenario.

The example in FIG. 3 demonstrates the accuracy of the new model in thepresence of proximity effects. The p.u.l. resistance and inductance areextracted for a differential line pair made of two copper wiresfeaturing 2 mm square cross-section and various separation distances d.FIG. 3 depicts the extracted resistance and inductance of theconfiguration as a function of frequency. The results obtained using theproposed model are compared with those published in C. R. Paul, Analysisof Mulliconduclor Transmission Lines, Ch. 3, John Wiley & Sons, Inc.,Toronto, CA, 1994 and D. DeZutter, et al., “Skin Effect Modeling Basedon a Differential Surface Admittance Operator,” IEEE Trans. MicrowaveTheory Tech., vol. 53, no. 8, pp. 2526-2538, August. 2005. Thefull-periphery EFIE solver yields dc resistance (independent from d asexpected) of 8.929 mΩ/m, which agrees with the dc resistance value in C.R. Paul, Analysis of Mulliconduclor Transmission Lines, Ch. 3, JohnWiley & Sons, Inc., Toronto, CA, 1994. The dc inductance was computed tobe 600.6 nH/m (599.5 nH/rn in C. R. Paul, Analysis of MulliconductorTransmission Lines, Ch. 3, John Wiley & Sons, Inc., Toronto, CA, 1994)for d=2 mm and 412.6 nH/m (413 nH/m in D. DeZutter, et al., “Skin EffectModeling Based on a Differential Surface Admittance Operator,” IEEETrans. Microwave Theory Tech., vol. 53, no. 8, pp. 2526-2538, August.2005) for d=0.5 mm.

Attention is now drawn to full-peripheral impedance for skin effectapproximation in a three-dimensional electric field integral equation.

Surface Electric Field Integral Equation And Conductor Loss Model

Let us consider a generic planar interconnect structure, depicted inFIG. 4, occupying volume V in the medium of L homogeneous dielectriclayers, which are infinitely extending in the xy-plane and exhibitinhomogeneity along the z-axis. The interconnect is made up of a verygood conducting material where conduction current is dominant anddisplacement current is negligible. The layers are characterized bycomplex permittivity {circumflex over (∈)}_(l)=∈_(l)+σ_(l)/(jω∈_(l)) andpermeability μ_(l) within intervals h_(l)<z<h_(l+1) where l=1, . . . , Land i=√{square root over (−1)}. In the above and throughout thefollowing derivations the time-harmonic field variation is assumed andsuppressed for brevity. The electric field in a multilayered medium isrelated to volumetric electric current density j through the vectorpotential A as

$\begin{matrix}{{E(r)} = {{E^{inc}(r)} - {\left( {{\omega} + {\frac{1}{{\omega}\hat{\varepsilon}\mu}{{\nabla\nabla} \cdot}}} \right){{A(r)}.}}}} & (13) \\{where} & \; \\{{A(r)} = {\int_{V}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {j\left( r^{\prime} \right)}}{{v^{\prime}}.}}}} & (14)\end{matrix}$

In above noted equation (14), G _(A) is the layered medium Greensfunction in standard form

G _(A) ={circumflex over (x)}{circumflex over (x)}G _(A,xx) +ŷŷG _(A,xx)+{circumflex over (z)}{circumflex over (z)}G _(A,zz) +{circumflex over(z)}{circumflex over (x)}G _(A,zx) +{circumflex over (z)}ŷG_(A,zx).  (15)

The electric field integral equation (EFIE) with respect to current jflowing in the conductor due to given excitation E^(inc) follows fromOhm's law E(r)=σ⁻¹j(r) enforced inside the conductor

$\begin{matrix}{{{\sigma^{- 1}{j(r)}} = {{E^{inc}(r)} - {\left( {{\omega} + {\frac{1}{{\omega}\hat{\varepsilon}\mu}{{\nabla\nabla} \cdot}}} \right){\int_{V}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {j\left( r^{\prime} \right)}}{v^{\prime}}}}}}},{r \in {V.}}} & (16)\end{matrix}$

Numerical solution of the volumetric integral equation (14) may be timeconsuming. Therefore, we use the surface impedance approximation toreduce the volumetric EFIE to the surface integral equation. Consideringthe rectangular cross-section of the conductors with local coordinatesystem introduced as shown in FIG. 4 we approximate dependence of thevolumetric current density j across the thickness of the conductoraccording to the skin-effect

$\begin{matrix}{{j(r)} \cong {\frac{\; {k_{\sigma}\left( {{{J\left( r^{t} \right)}^{{- }\; {k_{\sigma}{({T - {z{(r)}}})}}}} + {{J\left( r^{b} \right)}^{{- }\; k_{\sigma}{z{(r)}}}}} \right)}}{1 - ^{{- }\; k_{\sigma}T}} + \frac{\; {k_{\sigma}\left( {{{J\left( r^{l} \right)}^{{- }\; {k_{\sigma}{({x{(r)}})}}}} + {{J\left( r^{r} \right)}^{{- }\; {k_{\sigma}{({X - {x{(r)}}})}}}}} \right)}}{1 - ^{{- }\; k_{\sigma}W}}}} & (17)\end{matrix}$

where J(r^(t)), J(r^(b)), J(r^(l)), J(r^(r)) are the surface currentdensities the projection of radius-vector r the top, bottom, left, andright walls of the conductor, respectively, k_(σ)=√{square root over(ωμ_(0/)2)}(1−i) is the wavenumber of conductor. In equation (17) theconductor's thickness and its width at location r are denoted as T andW, respectively. On substitution of formula (17) into the integralequation we obtain

$\begin{matrix}{{\int_{V}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {j\left( r^{\prime} \right)}}{v^{\prime}}}} \cong {{\int_{S^{t}}{\int_{z^{\prime} = 0}^{T}{{{{\overset{\_}{G}}_{A}\left( {r,{r^{\prime} - {{\hat{n}\left( r^{\prime} \right)}z^{\prime}}}} \right)} \cdot \frac{{J\left( r^{\prime} \right)}^{{- }\; k_{\sigma}z^{\prime}}\; k_{\sigma}}{1 - ^{{- }\; k_{\sigma}T}}}{z^{\prime}}{s^{\prime}}}}} + {\int_{S^{b}}{\int_{z^{\prime} = 0}^{T}{{{{\overset{\_}{G}}_{A}\left( {r,{r^{\prime} + {{\hat{n}\left( r^{\prime} \right)}z^{\prime}}}} \right)} \cdot \frac{{J\left( r^{\prime} \right)}^{{- }\; k_{\sigma}z^{\prime}}\; k_{\sigma}}{1 - ^{{- }\; k_{\sigma}T}}}{z^{\prime}}{s^{\prime}}}}} + {\int_{S^{l}}{\int_{x^{\prime} = 0}^{W}{{{{\overset{\_}{G}}_{A}\left( {r,{r^{\prime} + {{\hat{n}\left( r^{\prime} \right)}x^{\prime}}}} \right)} \cdot \frac{{J\left( r^{\prime} \right)}^{{- }\; k_{\sigma}x^{\prime}}\; k_{\sigma}}{1 - ^{{- }\; k_{\sigma}W}}}{x^{\prime}}{s^{\prime}}}}} + {\int_{S^{r}}{\int_{x^{\prime} = 0}^{W}{{{{\overset{\_}{G}}_{A}\left( {r,{r^{\prime} - {{\hat{n}\left( r^{\prime} \right)}x^{\prime}}}} \right)} \cdot \frac{{J\left( r^{\prime} \right)}^{{- }\; k_{\sigma}x^{\prime}}\; k_{\sigma}}{1 - ^{{- }\; k_{\sigma}W}}}{x^{\prime}}{{s^{\prime}}.}}}}}} & (18)\end{matrix}$

where the source radius-vector r′ resides on the conductor surface S,i.e. r′∈S, S being the union of the top, bottom, left, and rightconductor walls S^(t), S^(b), S^(l), and S^(r) respectively. Assumingsmall thickness of conductor we can ignore the variation of the Green'sfunction across the conductor cross-section G _(A)(r,r′±{circumflex over(n)}(r′)z′)≅ G _(A)(r,r′) and G _(A)(r,r′±{circumflex over(n)}(r′)x′)≅G_(A)(r,r′) in equation (18). Such approximation allows usto evaluate analytically the integrals over conductor thickness T andwidth W in equation (18) and combine the four integral terms over thetop, bottom, left, and right surfaces, yielding

$\begin{matrix}{{\int_{V}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {j\left( r^{\prime} \right)}}{v^{\prime}}}} \cong {\int_{S}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {J\left( r^{\prime} \right)}}{{s^{\prime}}.}}}} & (19)\end{matrix}$

Thus, instead of the volume integral equation (16) we obtain the surfaceintegral equation with respect to the unknown surface current densities

$\begin{matrix}{{Z^{s}\left\{ {J\left( r^{\prime} \right)} \right\}} = {{\hat{n}(r)} \times {\quad{\left\lbrack {{E^{inc}(r)} - {\left( {{\omega} + {\frac{1}{{\omega}\hat{\varepsilon}\mu}{{\nabla\nabla} \cdot}}} \right){\int_{S}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {J\left( r^{\prime} \right)}}{s^{\prime}}}}}} \right\rbrack,r,{r^{\prime} \in {S.}}}}}} & (20)\end{matrix}$

In equation (20) Z^(s){J(r′)} denotes the surface impedance operatorrelating tangential electric field on the surface of conductor to thevalues of the surface current densities J(r′) at the correspondingpoints on the top, bottom, left, and right sides of the conductoraccording to E(r)=σ⁻¹j(r) and the relationship in equation (17) betweenthe volumetric and surface current densities

$\begin{matrix}{\quad\begin{matrix}{{{\hat{n}(r)} \times {E(r)}} = {Z^{s}\left\{ {J\left( r^{\prime} \right)} \right\}}} \\{= \left\lbrack {{E\left( {r \in S^{t}} \right)},{E\left( {r \in S^{b}} \right)},{E\left( {r \in S^{l}} \right)},{E\left( {r \in S^{r}} \right)}} \right\rbrack^{t_{p}}} \\{= {\frac{\; k_{\sigma}}{\sigma} \times}} \\{{\begin{bmatrix}\frac{1}{1 - ^{{- }\; k_{\sigma}T}} & \frac{^{{- }\; k_{\sigma}T}}{1 - ^{{- }\; k_{\sigma}T}} & \frac{^{{- }\; k_{\sigma}x}}{1 - ^{{- }\; k_{\sigma}W}} & \frac{^{{- }\; {k_{\sigma}{({W - x})}}}}{1 - ^{{- }\; k_{\sigma}W}} \\\frac{^{{- }\; k_{\sigma}T}}{1 - ^{{- }\; k_{\sigma}T}} & \frac{1}{1 - ^{{- }\; k_{\sigma}T}} & \frac{^{{- }\; k_{\sigma}x}}{1 - ^{{- }\; k_{\sigma}W}} & \frac{^{{- }\; {k_{\sigma}{({W - x})}}}}{1 - ^{{- }\; k_{\sigma}W}} \\\frac{^{{- }\; {k_{\sigma}{({T - z})}}}}{1 - ^{{- }\; k_{\sigma}T}} & \frac{^{{- }\; k_{\sigma}z}}{1 - ^{{- }\; k_{\sigma}T}} & \frac{1}{1 - ^{{- }\; k_{\sigma}W}} & \frac{^{{- }\; k_{\sigma}W}}{1 - ^{{- }\; k_{\sigma}W}} \\\frac{^{{- }\; {k_{\sigma}{({T - z})}}}}{1 - ^{{- }\; k_{\sigma}T}} & \frac{^{{- }\; k_{\sigma}z}}{1 - ^{{- }\; k_{\sigma}T}} & \frac{^{{- }\; k_{\sigma}W}}{1 - ^{{- }\; k_{\sigma}W}} & \frac{1}{1 - ^{{- }\; k_{\sigma}W}}\end{bmatrix} \cdot}} \\{\begin{bmatrix}{J\left( {{r^{\prime}(r)} \in S^{t}} \right)} \\{J\left( {{r^{\prime}(r)} \in S^{b}} \right)} \\{J\left( {{r^{\prime}(r)} \in S^{l}} \right)} \\{J\left( {{r^{\prime}(r)} \in S^{r}} \right)}\end{bmatrix}}\end{matrix}} & (21)\end{matrix}$

Superscript t_(p) in formula (21) denotes transposition.

Method of Moments Discretization of the Surface Impedance Operator

In order to make surface EFIE (8) amenable to Rao-Wilton-Glission (RWG)solution (as per S. M. Rao, D. R. Wilton, and A. W. Glisson,“Electromagnetic scattering by surfaces of arbitrary shapes,” IEEETrans. Antennas Propag., Vol. 30, pp. 409-418, May 1982 and K. A.Michalski and D. Zheng, “Electromagnetic scattering and radiation bysurfaces of arbitrary shape in layered media, Part I: Theory,” IEEETrans. Antennas Propag., Vol. 38, pp. 335-344, March 1990) theinterconnect surface is modeled by a triangulated plane extruded to itscorresponding thickness (FIG. 4). The discussion of the full peripherysurface impedance operator RWG discretization is detailed below.Considering a single conductor the traditional RWG discretization of theunknown current J in (20) over N basis functions ƒ is equivalentlyformulated as a discretization over the ramp-functions (half-RWGfunctions) situated at the top, bottom, left, and right walls ofconductor as follows

$\begin{matrix}{{{J(r)} \simeq {\sum\limits_{j = 1}^{N}{J_{j}{f_{j}(r)}}}} = {\sum\limits_{{w = t},b,l,r}{\sum\limits_{m = 1}^{M^{w}}{\sum\limits_{j = 1}^{3}{I_{j,m}^{w}{{R_{j,m}^{w}(r)}.}}}}}} & (22)\end{matrix}$

where index W denotes which wall the m^(th) triangle supporting rampR_(j.m) ^(w) is situated on and index j runs over three ramps on eachtriangle in according with FIG. 4.

The ramp-function discretization equation (22) of the unknown current Junder the surface impedance operator Z^(s) in equation (20) followed bytesting of the resultant electric field Z^(s){J} with ramp-functionR_(i,n) ^(t) in of top-wall triangle S_(n) ^(t) yields

R _(i,n) ^(t)(r),Z ^(s) {J}

=

R _(i,n) ^(t)(r),Z _(tt) ^(s) J(r _(m) ^(t)(r))+Z _(t,b) ^(s) J(r _(m)^(b)(r))+Z _(tl) ^(s)(r))J(r _(m) ^(l)(r))+Z _(tr) ^(s) J(r _(m)^(r)(r))

  (23)

where J(r_(m) ^(t)), J(r_(m) ^(b)), J(r_(m) ^(l)), J(r_(m) ^(r)), arethe current values at the corresponding points r_(m) ^(t), r_(m) ^(b),r_(m) ^(l), r_(m) ^(r), source triangles S_(m) ^(t), S_(m) ^(b), S_(m)^(l), S_(m) ^(r) at the top, bottom, left, and right walls,respectively, overlapping with triangle S_(n) ^(t) as shown in FIG. 4and FIG. 5. The current values on each source triangle are defined as asuperposition of three ramp-functions

$\begin{matrix}{{{{J\left( r_{m}^{t} \right)} = {\sum\limits_{j = 1}^{3}{I_{j,m}^{t}{R_{j,m}^{t}\left( r_{m}^{t} \right)}}}},{{J\left( {\overset{.}{r}}_{m}^{b} \right)} = {\sum\limits_{j = 1}^{3}{I_{j,m}^{b}{R_{j,m}^{b}\left( r_{m}^{b} \right)}}}}}{{{J\left( r_{m}^{l} \right)} = {\sum\limits_{j = 1}^{3}{I_{j,m}^{l}{R_{j,m}^{l}\left( r_{m}^{l} \right)}}}},{{J\left( r_{m}^{r} \right)} = {\sum\limits_{j = 1}^{3}{I_{j,m}^{r}{R_{j,m}^{r}\left( r_{m}^{r} \right)}}}},}} & (24)\end{matrix}$

and the corresponding values of the surface impedance operator are

$\begin{matrix}{{Z_{tt}^{s} = {\frac{\; k_{\sigma}}{\sigma}\frac{1}{1 - ^{{- }\; k_{\sigma}T}}}},{Z_{tb}^{s} = {\frac{\; k_{\sigma}}{\sigma}\frac{^{{- }\; k_{\sigma}T}}{1 - ^{{- }\; k_{\sigma}T}}}},{Z_{tl}^{s} = {\frac{\; k_{\sigma}}{\sigma}\frac{^{{- }\; k_{\sigma}x}}{1 - ^{{- }\; k_{\sigma}W}}}},{{Z_{tr}^{s}(r)} = {\frac{\; k_{\sigma}}{\sigma}{\frac{^{{- }\; {k_{\sigma}{({W - x})}}}}{1 - ^{{- }\; k_{\sigma}W}}.}}}} & (25)\end{matrix}$

In equation (24) I_(j,m) ^(w), w=t, b, l, r represent the sought unknowncoefficients in method of moments. From equations (23) through (25) wenotice that each, ith ramp R_(i,n) ^(t) on nth top wall, the observationtriangle S_(n) ^(t) has non-zero inner products

R_(i,n) ^(t), R_(j,m) ^(w)

with each of three ramps R_(j,m) ^(w), j=1, 2, 3, on overlappingtriangles S_(m) ^(w), w=t, b, l, r thus, yielding the following 12nonzero entries in each row of the discretized surface impedanceoperator

$\begin{matrix}{{I_{j,m}^{t}Z_{tt}^{s}{\langle{{R_{i,n}^{t}(r)},{R_{j,m}^{t}\left( {r_{m}^{t}(r)} \right)}}\rangle}} = {I_{j,m}^{t}\frac{\; k_{\sigma}}{\sigma}\frac{1}{1 - ^{{- }\; k_{\sigma}T}}{\int_{S_{n}^{t}}{{{R_{i,n}^{t}(r)} \cdot {R_{j,m}^{t}\left( {r_{m}^{t}(r)} \right)}}{S}}}}} & (26) \\{{I_{j,m}^{b}Z_{tb}^{s}{\langle{{R_{i,n}^{t}(r)},{R_{j,m}^{b}\left( {r_{m}^{b}(r)} \right)}}\rangle}} = {I_{j,m}^{b}\frac{\; k_{\sigma}}{\sigma}\frac{^{{- }\; k_{\sigma}T}}{1 - ^{{- }\; k_{\sigma}T}}{\int_{S_{n}^{t}}{{{R_{i,n}^{t}(r)} \cdot {R_{j,m}^{b}\left( {r_{m}^{b}(r)} \right)}}{S}}}}} & (27) \\{{I_{j,m}^{l}{\langle{{R_{i,n}^{t}(r)},{{Z_{tl}^{s}(r)}{R_{j,m}^{l}\left( {r_{m}^{l}(r)} \right)}}}\rangle}} = {I_{j,m}^{l}\frac{\; k_{\sigma}}{\sigma}\frac{1}{1 - ^{{- }\; k_{\sigma}W}}{\int_{S_{n}^{t}}{{{R_{i,n}^{t}(r)} \cdot {R_{j,m}^{l}\left( {r_{m}^{l}(r)} \right)}}^{{- }\; k_{\sigma}x}{S}}}}} & (28) \\{{I_{j,m}^{r}{\langle{{R_{i,n}^{t}(r)},{{Z_{tr}^{s}(r)}{R_{j,m}^{r}\left( {r_{m}^{r}(r)} \right)}}}\rangle}} = {I_{j,m}^{r}\frac{\; k_{\sigma}}{\sigma}\frac{1}{1 - ^{{- }\; k_{\sigma}W}}{\int_{S_{n}^{t}}{{{R_{i,n}^{t}(r)} \cdot {R_{j,m}^{r}\left( {r_{m}^{r}(r)} \right)}}^{{- }\; {k_{\sigma}{({W - x})}}}{S}}}}} & (29)\end{matrix}$

The integrals in equations (26) through (29) can be evaluatednumerically to arbitrary precision.

Since various modifications can be made in my invention as herein abovedescribed, and many apparently widely different embodiments of same madewithin the spirit and scope of the claims without department from suchspirit and scope, it is intended that all matter contained in theaccompanying specification shall be interpreted as illustrative only andnot in a limiting sense.

1. A method of modeling surface impedance of a conductor having acurrent flow, the method comprising: formulating a volumetric electricfield integral equation with respect to an unknown volumetric currentdensity of the current flow in the conductor; representing the unknownvolumetric current density in the volumetric electric field integralequation as a product of a current density of a peripheral surface ofthe conductor and an exponential factor describing a cross-sectionaldistribution of the current according to skin effect; adopting anapproximation of Green's function across a cross-section of theconductor; and reducing the volumetric electric field integral equationincluding the unknown volumetric current density representation to asurface integral equation using the approximation of Green's function.2. The method according to claim 1 including adopting an approximationof Green's function by fixing Green's function across the cross-sectionof the conductor.
 3. The method according to claim 1 for a conductorhaving a rectangular cross-section.
 4. The method according to claim 1including formulating the volumetric electric field integral equationwith respect to the unknown volumetric current density j_(z) in across-section of the conductor due to a vector of excitation V_(p.u.l.)by enforcing Ohm's law E_(z)(ρ)=σ⁻¹j_(z)(ρ) inside the conductor (ρ∈S)as follows:σ⁻¹j_(z)(ρ) + ωμ₀∫_(S)G(ρρ^(′))j_(z)(ρ^(′))S^(′) = −V_(p ⋅ u ⋅ l⋅)(ρ).5. The method according to claim 4 for a conductor which extends in az-direction and has a rectangular cross-section, the method includingapproximating the unknown volumetric current density, represented j_(z),across the conductor according to skin-effect of a plane-wave incidenton a conducting plane with infinite extension as follows:${j_{z}(\rho)} \cong {\frac{\; {k_{\sigma}\left( {{{J_{z}^{t}(y)}^{{- }\; {k_{\sigma}{({X - x})}}}} + {{J_{z}^{b}(y)}^{{- }\; k_{\sigma}x}}} \right)}}{1 - ^{{- }\; k_{\sigma}X}} + \frac{\; {k_{\sigma}\left( {{{J_{z}^{l}(x)}^{{- }\; k_{\sigma}y}} + {{J_{z}^{r}(x)}^{{- }\; {k_{\sigma}{({Y - y})}}}}} \right)}}{1 - ^{{- }\; k_{\sigma}Y}}}$where J_(z) ^(t)(y), J_(z) ^(b)(y), J_(z) ^(l)(x), J_(z) ^(r)(x), areunknown surface current densities at points of radius-vector ρprojections onto respective top, bottom, left, and right walls of theconductor.
 6. The method according to claim 5 including reducing thevolumetric electric field integral equation to a surface electric fieldintegral equation with respect to the current density of the peripheralsurface of the conductor by restricting an observation point ρ to aperiphery of the conductor as follows:Z^(σ){J_(z)(ρ)} + ωμ₀∫_(C)G(ρρ^(′))J_(z)(ρ^(′))c^(′) = −V_(p ⋅ u ⋅ l⋅)(ρ)7. The method according to claim 1 including combining unknown surfacecurrent densities over the top, bottom, left and right sides andevaluating integrals thereof over a conductor thickness T and conductorwidth W as follows:${\int_{V}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {j\left( r^{\prime} \right)}}{v^{\prime}}}} \cong {\int_{S}{{{{\overset{\_}{G}}_{A}\left( {r,r^{\prime}} \right)} \cdot {J\left( r^{\prime} \right)}}{{s^{\prime}}.}}}$8. The method according to claim 1 for a conductor having anon-rectangular polygonal cross-section.
 9. The method according toclaim 1 including formulating the volumetric electric field integralequation with respect to the unknown volumetric current density of thecurrent flow in the conductor for current flows along more than oneaxis.
 10. The method according to claim 1 including formulating thevolumetric electric field integral equation with respect to the unknownvolumetric current density of the current flow in the conductor forthree dimensional current flows.
 11. The method according claim 1including relating the unknown volumetric current density to unknownsurface current densities for each surface of the conductor in whicheach unknown surface current density is represented as amultidimensional vector.
 12. The method according to claim 1 includingreducing the volumetric electric field integral equation to the surfaceintegral equation in which a tangential electrical field at any point ona surface of the conductor is related to a surface current density atcorresponding points on all segments of a cross-sectional periphery ofthe conductor.
 13. The method according to claim 1 including reducingthe volumetric electric field integral equation to the surface integralequation in which a tangential electrical field at any point on asurface of the conductor is related to a tangential magnetic field atcorresponding points on all segments of a cross-sectional periphery ofthe conductor.
 14. A surface impedance model for a conductor having acurrent flow, the model comprising: a surface electric field integralequation resulting from a reduction of a volumetric electric fieldintegral equation in which an unknown cross-sectional volumetric currentof the volumetric electric field integral equation is approximated as aproduct of an exponential factor describing a cross-sectionaldistribution of the current according to skin effect and a currentdensity on a peripheral surface of the conductor. 15.-22. (canceled) 23.A method of implementing a surface impedance model of a conductorcomprising a surface electric field integral equation including asurface impedance term and an integral term resulting from a reductionof a volumetric electric field integral equation in which an unknowncross-sectional volumetric current of the volumetric electric fieldintegral equation is approximated as a product of an exponential factordescribing a cross-sectional distribution of the current according toskin effect and a current density on a peripheral surface of theconductor, the method including: discretizing the surface electric fieldintegral equation of a periphery of the conductor into a discrete formcomprising a sum of a sparse matrix corresponding to the surfaceimpedance term of the surface electric field integral equation and adense matrix corresponding to the integral term of the surface electricfield integral equation; and identifying for each discrete element ofthe discrete form, other discrete elements which are related via thesurface impedance model.
 24. The method according to claim 23 includingusing the surface impedance model in conjunction with a method of momentdiscretization scheme.
 25. The method according to claim 23 wherein thesurface impedance model comprises Green's function being fixed acrossthe cross-section of the conductor.
 26. The method according to claim 23for conductor having a rectangular cross-section.
 27. The methodaccording to claim 23 for a conductor having a non-rectangular polygonalcross-section.
 28. The method according to claim 23 wherein the surfaceimpedance model comprises the unknown cross-sectional volumetric currentof the volumetric electric field integral equation being approximatedfor current flows along more than one axis.
 29. The method according toclaim 23 wherein the surface impedance model comprises the volumetricelectric field integral equation being approximated for threedimensional current flows.
 30. The method according to claim 23 whereinthe surface impedance model comprises the unknown volumetric currentdensity being related to unknown surface current densities representedas multidimensional vectors for each surface of the conductor.
 31. Themethod according to claim 23 wherein the surface impedance modelcomprises a tangential electrical field at any point on a surface of theconductor being related to a surface current density at correspondingpoints on all segments of a cross-sectional periphery of the conductor.32. The method according to claim 23 wherein the surface impedance modelcomprises a tangential electrical field at any point on a surface of theconductor being related to a tangential magnetic field at correspondingpoints on all segments of a cross-sectional periphery of the conductor.33. The method according to claim 23 including representing the densematrix as follows:Z _(mn) ^(A) =iωμ ₀∫_(ΔC) _(n) G(ρ_(m)|σ′)dc′.